Ultrafilter Extensions and the Standard Translation
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The Nonstandard Theory of Topological Vector Spaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 THE NONSTANDARDTHEORY OF TOPOLOGICAL VECTOR SPACES BY C. WARD HENSON AND L. C. MOORE, JR. ABSTRACT. In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems ; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which tesults. Introduction. Let Ml be a set theoretical structure and let *JR be an enlarge- ment of M. Let (E, 0) be a topological vector space in M. §§1 and 2 of this paper are devoted to the elementary nonstandard theory of (F, 0). In particular, in §1 the concept of 0-finiteness for elements of *E is introduced and the nonstandard hull of (E, 0) (relative to *3R) is defined. §2 introduces the concept of 0-bounded- ness for elements of *E. In §5 the elementary nonstandard theory of locally convex spaces is developed by investigating the mapping in *JK which corresponds to a given pairing. In §§6 and 7 we make use of this theory by providing nonstandard treatments of two aspects of the existing standard theory. In §6, Luxemburg's characterization of the pre-nearstandard elements of *E for a normed space (E, p) is extended to Hausdorff locally convex spaces (E, 8). This characterization is used to prove the theorem of Grothendieck which gives a criterion for the completeness of a Hausdorff locally convex space. -
1. Introduction in a Topological Space, the Closure Is Characterized by the Limits of the Ultrafilters
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 08–37 THE ULTRAFILTER CLOSURE IN ZF GONC¸ALO GUTIERRES Abstract: It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with uX (A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈U]}. However, it is possible to built a topological space X for which uX 6= kX , but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {R}. Keywords: Ultrafilter Theorem, Ultrafilter Closure. AMS Subject Classification (2000): 03E25, 54A20. 1. Introduction In a topological space, the closure is characterized by the limits of the ultrafilters. Although, in the absence of the Axiom of Choice, this is not a fact anymore. -
SELECTIVE ULTRAFILTERS and Ω −→ (Ω) 1. Introduction Ramsey's
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 3067{3071 S 0002-9939(99)04835-2 Article electronically published on April 23, 1999 SELECTIVE ULTRAFILTERS AND ! ( ! ) ! −→ TODD EISWORTH (Communicated by Andreas R. Blass) Abstract. Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59{ 111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in L(R)is -Ramsey with respect to every selective ultrafilter . In this paper, we showU that the large cardinal assumption cannot be weakened.U 1. Introduction Ramsey's theorem [5] states that for any n !,iftheset[!]n of n-element sets of natural numbers is partitioned into finitely many∈ pieces, then there is an infinite set H ! such that [H]n is contained in a single piece of the partition. The set H is said⊆ to be homogeneous for the partition. We can attempt to generalize Ramsey's theorem to [!]!, the collection of infinite subsets of the natural numbers. Let ! ( ! ) ! abbreviate the statement \for every [!]! there is an infinite H ! such−→ that either [H]! or [H]! = ." UsingX⊆ the axiom of choice, it is easy⊆ to partition [!]! into two⊆X pieces in such∩X a way∅ that any two infinite subsets of ! differing by a single element lie in different pieces of the partition. Obviously such a partition can admit no infinite homogeneous set. Thus under the axiom of choice, ! ( ! ) ! is false. Having been stymied in this naive−→ attempt at generalization, we have several obvious ways to proceed. A natural project would be to attempt to find classes of partitions of [!]! that do admit infinite homogeneous sets. -
Logical Compactness and Constraint Satisfaction Problems
Logical Methods in Computer Science Vol. 13(1:1)2017, pp. 1–11 Submitted Jan. 21, 2016 https://lmcs.episciences.org/ Published Jan. 23, 2017 LOGICAL COMPACTNESS AND CONSTRAINT SATISFACTION PROBLEMS DANNY RORABAUGH a, CLAUDE TARDIF b, AND DAVID WEHLAU c a Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada b,c Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada Abstract. We investigate a correspondence between the complexity hierarchy of con- straint satisfaction problems and a hierarchy of logical compactness hypotheses for finite relational structures. It seems that the harder a constraint satisfaction problem is, the stronger the corresponding compactness hypothesis is. At the top level, the NP-complete constraint satisfaction problems correspond to compactness hypotheses that are equiva- lent to the ultrafilter axiom in all the cases we have investigated. At the bottom level, the simplest constraint satisfaction problems correspond to compactness hypotheses that are readily provable from the axioms of Zermelo and Fraenkel. 1. Introduction A relational structure A is said to be compact if for any structure B of the same type, B admits a homomorphism to A whenever every finite substructure B′ of B admits a ho- momorphism to A. The compactness theorem of logic implies that every finite structure is compact. However, the compactness theorem is equivalent to the ultrafilter axiom, a consequence of the axiom of choice that is not provable from the axioms of Zermelo and Fraenkel. In this paper, we restrict our attention to the case where A is finite, but we do not assume the ultrafilter axiom. -
Topological Vector Spaces
An introduction to some aspects of functional analysis, 3: Topological vector spaces Stephen Semmes Rice University Abstract In these notes, we give an overview of some aspects of topological vector spaces, including the use of nets and filters. Contents 1 Basic notions 3 2 Translations and dilations 4 3 Separation conditions 4 4 Bounded sets 6 5 Norms 7 6 Lp Spaces 8 7 Balanced sets 10 8 The absorbing property 11 9 Seminorms 11 10 An example 13 11 Local convexity 13 12 Metrizability 14 13 Continuous linear functionals 16 14 The Hahn–Banach theorem 17 15 Weak topologies 18 1 16 The weak∗ topology 19 17 Weak∗ compactness 21 18 Uniform boundedness 22 19 Totally bounded sets 23 20 Another example 25 21 Variations 27 22 Continuous functions 28 23 Nets 29 24 Cauchy nets 30 25 Completeness 32 26 Filters 34 27 Cauchy filters 35 28 Compactness 37 29 Ultrafilters 38 30 A technical point 40 31 Dual spaces 42 32 Bounded linear mappings 43 33 Bounded linear functionals 44 34 The dual norm 45 35 The second dual 46 36 Bounded sequences 47 37 Continuous extensions 48 38 Sublinear functions 49 39 Hahn–Banach, revisited 50 40 Convex cones 51 2 References 52 1 Basic notions Let V be a vector space over the real or complex numbers, and suppose that V is also equipped with a topological structure. In order for V to be a topological vector space, we ask that the topological and vector spaces structures on V be compatible with each other, in the sense that the vector space operations be continuous mappings. -
Ultrafilters and Tychonoff's Theorem
Ultrafilters and Tychonoff's Theorem (October, 2015 version) G. Eric Moorhouse, University of Wyoming Tychonoff's Theorem, often cited as one of the cornerstone results in general topol- ogy, states that an arbitrary product of compact topological spaces is compact. In the 1930's, Tychonoff published a proof of this result in the special case of [0; 1]A, also stating that the general case could be proved by similar methods. The first proof in the gen- eral case was published by Cechˇ in 1937. All proofs rely on the Axiom of Choice in one of its equivalent forms (such as Zorn's Lemma or the Well-Ordering Principle). In fact there are several proofs available of the equiv- alence of Tychonoff's Theorem and the Axiom of Choice|from either one we can obtain the other. Here we provide one of the most popular proofs available for Ty- chonoff's Theorem, using ultrafilters. We see this approach as an opportunity to learn also about the larger role of ultra- filters in topology and in the foundations of mathematics, including non-standard analysis. As an appendix, we also include an outline of an alternative proof of Tychonoff's Theorem using a transfinite induction, using the two-factor case X × Y (previously done in class) for the inductive step. This approach is found in the exercises of the Munkres textbook. The proof of existence of nonprincipal ultrafilters was first published by Tarski in 1930; but the concept of ultrafilters is attributed to H. Cartan. 1. Compactness and the Finite Intersection Property Before proceeding, let us recall that a collection of sets S has the finite intersection property if S1 \···\ Sn 6= ? for all S1;:::;Sn 2 S, n > 0. -
Chapter 6 ULTRA-METHODS in METRIC FIXED POINT THEORY
Chapter 6 ULTRA-METHODS IN METRIC FIXED POINT THEORY M. A. Khamsi Department of Mathematical Sciences and Computer Science The University of Texas at El Paso El Paso, Texas 79968, USA B. Sims Mathematics, School of Mathematical and Physzcal Sciences The University of Newcastle NSW 2308, Australia 1. Introduction Over the last two decades ultrapower techniques have become major tools for the development and understanding of metric fixed point theory. In this short chapter we develop the Banach space ultrapower and initiate its use in studying the weak fixed point property for nonexpansive mappings. For a more extensive and detailed treatment than is given here the reader is referred to [l]and [21]. 2. Ultrapowers of Banach spaces Throughout the chapter I will denote an index set, usually the natural numbers N,for most situations in metric fixed point theory. Definition 2.1 A filter on I is a nonempty family of subsets F C 2' satisfying (i) F is closed under taking supersets. That is, A E F and A c B c I ==+ B E F. (ii) F is closed under finite intersections: A, B E F ==+ A n B E F. Examples. (1) The power set of I, 2', defines a filter. (2) The Fr6chet filter {Ac I : I\A is finite } 177 WA. Kirk and B. Sims (eds.),Handbook of Metric Fixed Point Theory, 177-199. O 2001 Kluwer Academic Publishers. Printed in the Netherlands. (3) For io E I, Fi0 := {Ac I : io E A}. Filt,ers of the form Fto for some io E I are t,ermed trivial, or non-free filters. -
Ultrafilters and Topology
Leiden University Bachelor project Mathematics Ultrafilters and Topology Author: Supervisor: Marius Stekelenburg Dr. Owen Biesel June 13, 2014 Ultrafilters and topology Marius Stekelenburg June 13, 2014 Contents 1 Introduction 2 2 Ultrafilters and topology 3 2.1 Fundamental matters . .3 2.2 Convergence and topological properties . .4 2.3 Application: Tychonov’s theorem . .7 3 A categorical approach 9 3.1 Monads and algebras . .9 3.2 The ultrafilter monad . 12 3.3 Manes’ theorem . 14 4 Barr’s theorem 18 5 The Stone-Čech compactification 21 5.1 Stone-Čech compactification of discrete spaces . 21 5.2 Stone-Čech compactification of arbitrary spaces . 23 1 1 Introduction In this thesis, we will explore a simple, but rich concept, called ultrafilters. This concept has utilities in many areas, such as model theory, logic, and topology (this can be read in [2]). As the title suggests, this thesis has its focus on the latter area, topology. On topological spaces, ultrafilters can converge. Many topological prop- erties can be formulated in terms of this convergence. They work in is similar fashion to the ordinary sequence convergence, with one notable difference: it works for every topological space. This comes with a nice bonus: Tychonov’s theorem, a theorem that is otherwise hard to prove! The main result of this thesis will a reconstruction of the category of compact Hausdorff spaces. Originally proven by Ernest Manes, this con- struction enables a very different way of creating topological spaces, what will come in handy last chapter. When finding a major result, one could do two things with it: general- izing or finding an application. -
ZF, Choice, Zorn, Ordinals, Ultrafilters
ZF, Choice, Zorn, Ordinals, Ultrafilters Rich Schwartz July 16, 2021 1 Introduction There are various foundations for what you could call everyday mathematics. These foundations usually come as a list of axioms describing the allowable kinds of mathematical reasoning and constructions. The most popular foun- dational axioms are the Zermelo-Frankel Axioms. Most of them sound com- pletely obvious. However, the ZF axioms are not really stand-alone axioms. In order to understand and use them, you need language and logic. Which comes first, the language, the logic, or the axioms? I don't know. In practice, when you do mathematics, you freely use all three things in ways that are not so easy to disentangle. There are some additional axioms which do not follow from the ZF axioms (and language and logic). The most famous is the Axiom of Choice, an axiom which has many reformulations { e.g. Zorn's Lemma. One purpose of these notes is to discuss the ZF axioms, with a view towards putting the Axiom of Choice in context. The main purpose is to show the equivalence of the Axiom of Choice, Zorn's Lemma, and the Well-Ordering Principle, given ZF. Finally, the last purpose is to derive some nice applications of these axioms { e.g., the existence of nonprinciple ultrafilters. I got many of the details about the ZF axioms from Wikipedia and from An introduction to Set Theory by William A.R. Weiss, though these notes are not a perfectly faithful account of what is written there. In particular, the arguments in the notes do not \go all the way back to the axioms", though of course they could be re-written (at length) to do so. -
Nets and Filters (Are Better Than Sequences)
Nets and filters (are better than sequences) Contents 1 Motivation 2 2 More implications we wish would reverse2 3 Nets 4 4 Subnets 6 5 Filters 9 6 The connection between nets and filters 12 7 The payoff 14 8 Filling in a gap from first year calculus 15 c 2018{ Ivan Khatchatourian 1 Nets and filters (are better than sequences) 2. More implications we wish would reverse 1 Motivation In the Section 5 of the lecture notes we saw the following two results: Theorem 1.1. Let (X; T ) be a Hausdorff space. Then every sequence in X converges to at most one point. Proposition 1.2. Let (X; T ) be a topological space, let A ⊆ X, and let a 2 X. If there is a sequence of points in A that converges to a, then a 2 A. We discussed how both of these implications really feel like they should reverse, but unfor- tunately neither of them do. In both cases, additionally assuming the topological space is first countable allows the implications to reverse. This is fine, but it still feels like sequences are not quite powerful enough to capture the ideas we want to capture. The ideal solution to this problem is to define a more general object than a sequence|called a net|and talk about net convergence. That is what we will do in this note. We will also define a type of object called a filter and show that filters also furnish us with a type of convergence which turns out to be equivalent to net convergence in all ways. -
[Math.LO] 22 Dec 2012 Filters and Ultrafilters in Real Analysis
Filters and Ultrafilters in Real Analysis Max Garcia Mathematics Department California Polytechnic State University San Luis Obispo, California 93407, USA E-mail: [email protected] arXiv:1212.5740v1 [math.LO] 22 Dec 2012 Abstract We study free filters and their maximal extensions on the set of natural numbers. We characterize the limit of a sequence of real numbers in terms of the Fr´echet filter, which involves only one quantifier as opposed to the three non-commuting quantifiers in the usual definition. We construct the field of real non-standard numbers and study their properties. We characterize the limit of a sequence of real numbers in terms of non-standard numbers which only requires a single quantifier as well. We are trying to make the point that the involvement of filters and/or non-standard numbers leads to a reduction in the number of quantifiers and hence, simplification, compared to the more traditional ε, δ-definition of limits in real analysis. Keywords and phrases: Limit, sequential approach to real analysis, fil- ter, Fr´echet filter, ultrafilter, non-standard analysis, reduction of quantifiers, infinitesimals, monad, internal sets. AMS Subject Classification: 03C10, 03C20, 03C50, 03H05, 12L10, 26E35, 26A03, 26A06, 30G06Keywords and phrases: Limit, sequential approach to real analysis, filter, Fr´echet filter, ultrafilter, non-standard analysis, re- duction of quantifiers, infinitesimals, monad, internal sets. AMS Subject Classification: 03C10, 03C20, 03C50, 03H05, 12L10, 26E35, 26A03, 26A06, 30G06 Contents Introduction............................... 1 1 Filters, Free Filters and Ultrafilters 3 1.1 Filters and Ultrafilters . 3 1.2 Existence of Free Ultrafilters . 5 1.3 Characterization of the Ultrafilter . -
Ultrafilters with Property
Ultrafilters with property (s) Arnold W. Miller1 Abstract A set X ⊆ 2ω has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2ω there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forc- ing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that as- suming CH (or more generally MA) there exists an ultrafilter U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of ω. It is a well known classical result due to Sierpinski (see [1]) that a non- principal ultrafilter U on ω when considered as a subset of P (ω) = 2ω cannot have the property of Baire or be Lebesgue measurable. Here we identify 2ω and P (ω) by identifying a subset of ω with its characteristic function. An- other very weak regularity property is property (s) of Marczewski (see Miller [7]). A set of reals X ⊆ 2ω has property (s) iff for every perfect set P there exists a perfect subset Q ⊆ P such that either Q ⊆ X or Q ∩ X = ∅. Here by perfect we mean homeomorphic to 2ω. It is natural to ask: Question.